A tour agency sells cruise packages to Vietnam, Maldives and Taiwan at $1299, $1398 and $2538 respectively. The table below shows the number of customers who have signed up for the respective cruise packages via the tour agency from January to June and from July to December. The number of people who signed up for cruise packages to Vietnam, Maldives and Taiwan can be represented by the matrix $P=\begin{pmatrix}27 & 24\\23 & 29\\19 & 22\end{pmatrix}$

1. Write the price matrix in the correct order and dimensions for $C=AP$.

Since $P$ is a $3\times 2$ matrix with rows in the order Vietnam, Maldives, Taiwan, and $C$ must give total earnings for each period, $A$ must be a $1\times 3$ row matrix containing the corresponding prices:
$A=\begin{pmatrix}1299 & 1398 & 2538\end{pmatrix}.$

1. Set up the matrix multiplication $C=AP$ using the price matrix and the customer matrix.

$C=AP$
$=\begin{pmatrix}1299 & 1398 & 2538\end{pmatrix}\begin{pmatrix}27 & 24\\23 & 29\\19 & 22\end{pmatrix}$

2. Multiply corresponding entries and add to find each period's total earnings.

$C=\begin{pmatrix}(1299)(27)+(1398)(23)+(2538)(19) & (1299)(24)+(1398)(29)+(2538)(22)\end{pmatrix}$
$=\begin{pmatrix}35073+32154+48222 & 31176+40542+55836\end{pmatrix}$
$=\begin{pmatrix}115449 & 127554\end{pmatrix}$

1. Set up the matrix multiplication $M=CR$ using the given column matrix $R$.

$M=CR=\begin{pmatrix}118\,911 & 127\,584\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix}$

2. Multiply corresponding entries and add to obtain the single element of $M$.

$M=\begin{pmatrix}118\,911(1)+127\,584(1)\end{pmatrix}$

$M=\begin{pmatrix}246\,495\end{pmatrix}$

1. Interpret the single element of $M$ in the context of the question.

Since $M=CR=\begin{pmatrix}118\,911 & 127\,584\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix},$ the two period earnings are added together. Hence, the element of $M$ represents the total earnings for both periods combined, from January to December.